INTRODUCTION OF THE PHONE-CONCEPT INTO POLE FIGURE INVERSION USING THE ITERATIVE SERIES EXPANSION METHOD

The phone-concept as it is used in the various kinds of probabilistic methods can easily be applied to the iterative series expansion method for quantitative texture analysis. Only slight modifications of the existing routines are necessary. The advantages of this concept are demonstrated by a mathematical and an experimental example.


INTRODUCTION
It is the major drawback of the series expansion method that after pole figure inversion, negative pole densities may appear in recalculated pole figures. In the case of incomplete pole figures, this may happen in the unmeasured range of the measured pole figures (Dahms, 1987). If only a small number of pole figures is measured, strong negative intensities may occur in recalculated unmeasured pole figures (Dahms & Bunge, 1988a). Also the iterative series expansion method, the so-called "positivity method" (Dahms & Bunge, 1989), does not necessarily result in completely positive recalculated pole figures (Dahms, 1992). If there are negative pole densities in recalculated pole figure, being physically impossible, it is impossible to obtain a totally positive complete ODF. It is the aim of this paper to show that a slight modification of the positivity method leads to a considerable improvement of the pole figure inversion with respect to positivity of recalculated pole figures.

GENERAL CONSIDERATIONS
In the positivity method (Dahms & Bunge, 1989), after a first approximation of the series expansion coefficients C 'v, the unmeasured range of the measured pole figures and additional unmeasured pole figures are calculated. Any of these pole densities below zero is set to zero, and using these corrected pole figures, a second approximation of the Ct v is calculated. Using this loop, an iterative procedure is defined, which is the "even" part of the positivity method. The iteration is stopped, if no further negative values occur, no further improvement of the Cf is obtained or if a time barrier is reached. 169 In the case of textures with an isotropic component r or "phone" (Matthies, 1984), this procedure will obviously not lead to a correct pole figure inversion if just a few pole figures are measured. In the best case, unmeasured recalculated pole figures will contain regions with zero pole density if the first approximation of the C leads to negative pole densities.
The situation is improved if not only negative pole densities are set to zero but densities less than r are set to r. In general, if an infinite number of additional pole figures is introduced, i.e., the whole inverse pole figure is covered, the original texture will be exactly reproduced. In fact, the boundary condition "no pole density less than r" is a higher constraint than the positivity condition.
In practice, it is not known whether there is a phone present or not. Also, if the "even" ODF has an isotropic component, i.e., the absolute minimum of all pole figures is greater than zero, just the range of ambiguity of the complete ODF is touched (Matthies, 1979). In case of a limited number of measured pole figures, the problem is even worse. The minimum of all pole figures is a value between zero and the minimum pole density of the measured pole figures.
In the present variant of the iterative series expansion method, the value r is entered manually in the routine and chosen according to the experience of the operator. As a criterion for a proper choice of r, the "simplicity" of the recalculated pole figures is chosen, i.e., a texture should be interpreted by a small number of texture components. Here, some relation to the component method (Helming and Eschner, 1990) can be seen.

EXAMPLES
In order to show the versality of the suggested generalization of the positivity method, two examples will be presented. The first one is a synthetic (001)[100Jtexture with a gaussian spread of 12.5 and an isotropic component of r =0.5 assuming orthorhombic crystal and sample symmetries, see Dahms and Bunge (1988b). The second one is that of natural quartz as published by Braun et al. (1991).

Synthetic texture
As an extreme case, only the three (100), (010) and (001)  For pole figure inversion, the (111), 3(110), 6{210), 3(311), and 6{321) pole figures were introduced in the calculation as described by Dahms and Bunge (1989). Hence, 19 additional pole figures were checked for positivity during the iterative procedure. The first calculation was done with only positivity assumed, the second calculation incorporated the isotropic component r 0.5. After pole figure inversion, the general axis distribution function (Bunge, 1982) was calculated in steps of 5 for all crystal and sample directions, i.e., for each crystal direction, a complete pole figure was calculated. In Figure 1, the minimum value of each individual pole figure is represented in inverse pole figures.  Figure  2b) are shown. There is no serious difference, except the minimum values which were raised from -0.12 to 0.21. Additionally, most of the smaller peaks taking place for r--0 disappear in the case of r 0.5. Obviously, the introduction of a positive minimum value acts as a tension towards real positivity and smoothness.
It is clear from these results that an over-estimation of r does not lead to minimum values in pole figures very close to r. If odd-order coefficients are determined using the positivity method (Dahms & Bunge, 1989), the minimum value of the complete ODF remains less than -1 for the case of r 0, i.e., the negative values in the pole figures are transferred to the ODF. For r 0.5, a minimum value of -0.5 is reached after 10 iterative cycles, see Figure 3. By separate calculations (Dahms, 1991), it has been shown that this minimum is due to series truncation effects. The difference between minimum values of about 0.5 is in the range of the assumed isotropic texture component r, and it is also in the range of the raised minimum of the (001) pole figure.
Here, we have a typical example, how a slight improvement of the "even" ODF leads to a considerable improvement of the complete ODF as it has been shown by Wagner et al. (1990).